Compared to the contraction factor \(\sigma =\max \big \lbrace 1-\frac{2\mu L\gamma }{\mu +L},1-\frac{\gamma \tau \rho }{2}\big \rbrace \)  for the unbiased compressors in theo1, theo4 shows that the linear convergence rate for biased compressors has the same dependence on \(\gamma ,\tau ,\mu ,L\)  and \(\rho \)  if \(\delta \)  is small, but reduces faster as \(\delta \)  increases. Fortunately, a moderate number of encoding bits often leads to a very small \(\delta \)  since it typically decreases exponentially in the encoding length as shown in exapcompressorREF . Moreover, the round-off error caused by finite-precision is a consequence of biased compression with a small \(\delta \) . For instance, \(\delta <1.2\times 10^{-7}\)  when using 32-bit floating-point format for \(|\mathbf {x}|>2^{-126}\)  [1]}, [2]}.
